We discuss a quartic eigenvalue problem arising in the context of an optical waveguiding problem involving atomically thick 2D materials. The waveguide configuration we consider consists of a gradient-index (spatially dependent) dielectric equipped with conducting interior interfaces. This leads to a quartic eigenvalue problem with mixed transverse electric and transverse magnetic modes, and strongly coupled electric and magnetic fields. We derive a weak formulation of the quartic eigenvalue problem and introduce a numerical solver based on a quadratification approach in which the quartic eigenvalue problem is transformed to a spectrally equivalent companion problem. We verify our numerical framework against analytical solutions for prototypical geometries. As a practical example, we demonstrate how an improved quality factor (defined by the ratio of the real and the imaginary part of the computed eigenvalues) can be obtained for a family of gradient-index host materials with internal conducting interfaces. We outline how this result lays the groundwork for solving related shape optimization problems.
Bibliographical noteFunding Information:
The authors were supported in part by ARO MURI Award W911NF-14-1-0247.The first and third author's research was supported in part by NSF Awards DMS-1819220 and DMS-1906129.This author's research was supported in part by NSF Award DMS-1912847.
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- Guided mode
- Nonlinear eigenvalue problem
- Quartic eigenvalue problem
- Surface plasmon polariton
- Time-harmonic Maxwell's equations